Jacob Bigeleisen

Vapor Pressures of 36Ar and 40Ar. Intermolecular Forces in Solid and Liquid Argon

The vapor pressures of 36Ar and 40Ar have been measured in the temperature range 62–102°K. Measurements in the liquid agree quantitatively with the previous measurements of Clusius et al. which cover the limited range 84–88°K. A significant difference is found between our equilibrium measurements, which have a precision of 0.5% in the logarithm of the vapor pressure ratio, and dynamic measurements of the isotope fractionation in solid–vapor systems. The reduced partition functions of the solid and liquid are calculated from the observed vapor pressure ratios. The reduced partition ratio of the solid can be represented by the equation T2ln(fs / fg) = 63.39 ± 0.34 − (0.124 ± 0.005)T − (95 / 63)(63.4 − 0.124T)2 / T2. The experimental values of ln(fs / fg) are in good agreement with calculations from the improved self‐consistent phonon theory of an anharmonic lattice with a 13–6 potential. Since the quantum effect is small, 〈∇2U〉 is obtained directly from the reduced partition functions. In the liquid range t...

Liquid‐Vapor Argon Isotope Fractionation from the Triple Point to the Critical Point. Mean Laplacian of the Intermolecular Potential in Liquid Argon

The statistical thermodynamic treatment of the equilibrium between a nonideal liquid mixture of isotopes and a vapor phase is extended to include isotope effects on the equation of state of the gas. The result is particularly simple when the isotopic partition functions in a given phase are compared at the same molar volume. The isotope fractionation factor α for 36Ar/40Ar between liquid and vapor has been measured from the triple point to the critical temperature. The results are compared with previous vapor pressure data, which cover the range 84–102°K. Although the agreement is within twice the statistical scatter of the present data, the present results for the lnα are systematically 5% lower than calculations from vapor pressure data. It is shown that T2 lnα is a linear function of (ρc−ρg), the density difference between the liquid and vapor, in the range 84–120°K. The fractionation factor approaches zero at the critical temperature with a nonclassical critical index equal to 0.42±0.02.〈∇2 Uc〉/ρc in ...

Vapor Pressures of Isotopic Krypton Mixtures. Intermolecular Forces in Solid and Liquid Krypton

The vapor pressures of normal krypton and a krypton sample containing 50% 80Kr have been measured from 94 to 130°K. New vapor pressure equations are derived for solid and liquid normal krypton. The triple‐point pressures and (temperatures) of 80Kr and 84Kr are 547.565 (115.77) and 547.502 (115.78) int. mm Hg (°K), respectively. The triple‐point pressure of normal krypton is 547.52 ± 0.10 int. mm Hg. The reduced partition functions, fc/fg, of 84Kr/80Kr have been derived for the solid and liquid from the vapor pressure data. The solid data lead to a θ0 (D) = 71.8°K in excellent agreement with heat capacity data. The change in ln (fc/fg) on melting is found to be 24% in good agreement with previous results on neon and argon. V 〈 ∇2 U 〉 is derived from the experimental values of ln (fc/fg) in the liquid. The repulsive potential in liquid krypton is less steep than that in neon or argon. Calculations of V 〈 ∇2 U 〉 from neutron scattering measurements show that the exp‐6 potential to be superior to the LJ (12, ...

Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic carbon dioxide molecules

The vapor pressures of samples of CO2 enriched in 13C and 18O have been compared with normal CO2 over the temperature range 154−217°K. Over the entire temperature range the order of the vapor pressures is P13CO2 > P12CO2 > PC18O2. The results are explained by the small shift in the internal vibrations, which dominate the carbon isotope effect, and hindered rotation in the solid and liquid. While the isotope effect on melting is small for carbon substitution, it is found to be large for oxygen substitution. Hindered rotation is the major contribution to the oxygen isotope effect in the solid, while the hindered translation is the most important contribution to the oxygen isotope effect in the liquid. Model calculations of the logarithm of the reduced partition function ratios of the solid and liquid, ln(fc/fg), are made in an anharmonic version of the Stern−Van Hook−Wolfsberg cell model. These calculations give good agreement with the experimentally determined ln(fc/fg) values for the solid, liquid−vapor i...

Isotope chemistry and molecular structure. Deviations from the first rule of the mean

The method of the finite orthogonal polynomial expansions of the Bigeleisen–Mayer function is used to study deviations from the first rule of the mean of isotope chemistry. It is shown by sample calculations for hydrogen isotope substitution in water, methane, and ethylene and carbon isotope substitution in ethylene that the second order corrections, (h//kT)4Σδλ2i account for all but ±0.001 to ±0.01 of the natural logarithm of the symmetry corrected exact harmonic disproportionation constant at 300 °K. The structure of the second order corrections have been related to the molecular structure through the use of internal coordinates. It is shown that subrules local to sets of equivalent coordinates exist in addition to the sum rules formed over the common symmetry classes or the totality of molecular eigenvalues. The local subrules are conveniently expressed in matrix form, D matrix. The latter can be used as a basis for the derivation of the sum rules. The first and second order D matrices are derived for ...

Isotope chemistry and molecular structure. Total deuterium isotope effects

A qualitative discussion is given on the relationship between the magnitude of isotope effects in polyatomic systems and bending and stretching force constants. This is followed by a general method for the calculation of the isotope effect which can be ascribed to each force constant in the general quadratic potential energy for the vibration of a polyatomic molecule. The method is applied to deuterium isotope effects in the molecules H2O, CH2O, CH4, C2H4, C2H6, and C6H6 at and above 300°K. It is found that deuterium isotope effects associated with individual internal valence force constants are nearly additive at room temperature. Strict additivity holds in the high temperature limit. Bond stretching force constants account for at least 65% of the total deuterium−to−protium reduced partition function ratio at 300°K and become even more dominant at higher temperatures. The origin of the relative importance of stretching and bending forces as a function of temperature is found in the first two terms of the...

Correlation of Isotope Effects with Molecular Forces. II. Triatomic Molecules

The theory of the relationship between the reduced partition function ratio of isotopic molecules and molecular force constants in triatomic molecules is developed through the application of finite orthogonal polynomials. Detailed calculations are presented for 13C and 18O substitution in CO2 at 500 and 1000°K, and 2H and 18O substitution in H2O at 1000 and 2000°K. The quantum correction of order j is jth order in diagonal force constant elements, but essentially linear in off‐diagonal force constant elements. Stretch‐bend interaction force constants are typically of increasing importance to the higher quantum corrections. In some cases, such as 18O substitution in water, the stretch‐bend interaction constant is an order of magnitude more important than the stretching force constant to all quantum corrections other than the simple first quantum correction. The quantum corrections are analyzed into contributions corresponding to the statistical mechanics of a single oscillating particle and coupling terms ...

Calculation of the mean force constants of the rare gases and the rectilinear law of mean force

The mean energies 〈U〉 and the mean force constants 〈∇2U〉 have been calculated for liquid argon and liquid krypton using the WCA perturbation theory, and for gaseous argon and krypton along the coexistence line by solution of the PY equation for the radial distribution function. Calculations have been made for the Lennard‐Jones, Barker–Henderson, and Maitland–Smith potentials. There is little difference in the values of 〈U〉 and 〈∇2U〉 calculated for the three potentials. The calculated values are in good agreement with experimental data. Correlation of the calculated values of 〈∇2U〉 for the liquid and gas leads to an empirical relationship between 〈∇2U〉 and the density of the phase along the coexistence line. When the latter is combined with the law of rectilinear diameters of the density, one obtains the rectilinear law of mean force, which is in good agreement with experimental data on argon. It is shown that the scaling exponent for the mean force constant 〈∇2U〉l‐〈∇2U〉g is larger than the scaling exponen...

Approximation of isotopic reduced partition function ratios by the WINIMAX polynomial perturbation method (WIMPER)

Diagonal H, F, and G matrix approximations to the moments of the eigenvalues are studied for a number of polyatomic molecules. For molecules containing hydrogen atoms the diagonal element approximations to each of the first six moments do not differ from the true values by more than 15%. The results further substantiate the simplified method of analysis of the isotope chemistry of hydrogen in terms of stretching and bending forces. It is shown that the coupling of bending and stretching coordinates makes a significant contribution to the motion of ’’central’’ atoms in a molecule. Corrections to the reduced partition function ratios of isotopic molecules corresponding to uncoupled internal coordinates, the diagonal element approximation, are developed through the use of the WINIMAX polynomials (WIMPER). The magnitude of the corrections are related directly to the off‐diagonal F and G matrix elements. Sample calculations show that the diagonal element approximations lead to values of ln (s/s′) f for deuteri...

Generalized finite polynomial approximation (WINIMAX) to the reduced partition function of isotopic molecules

The MINIMAX finite polynomial approximation to an arbitrary function has been generalized to include a weighting function (WINIMAX). It is suggested that an exponential is a reasonable weighting function for the logarithm of the reduced partition function of a harmonic oscillator. Comparison of the error function for finite orthogonal polynomial (FOP), MINIMAX, and WINIMAX expansions of the logarithm of the reduced vibrational partition function show WINIMAX to be the best of the three approximations. A condensed table of WINIMAX coefficients is presented. The FOP, MINIMAX, and WINIMAX approximations are compared with exact calculations of the logarithm of the reduced partition function ratios for isotopic substitution in H2O, CH4, CH2O, C2H4, and C2H6 at 300 °K. Both deuterium and heavy atom isotope substitution are studied. Except for a third order expansion involving deuterium substitution, the WINIMAX method is superior to FOP and MINIMAX. At the level of a second order expansion WINIMAX approximation...